Non-resonance with one-sided superlinear growth for indefinite planar systems via rotation numbers

Abstract

<abstract><p>We consider the non-resonance with one-sided superlinear growth conditions for the indefinite planar system $ z' = f(t, z) $ from a rotation number viewpoint, and obtain the existence of $ 2\pi $-periodic solutions by applying a rotation number approach together with the Poincaré-Bohl theorem. We allow that the angular velocity of solutions of $ z' = f(t, z) $ is controlled by the angular velocity of solutions of two positively homogeneous and oddly symmetric systems $ z' = L_i(t, z), i = 1, 2 $ on the left half-plane, which have rotation numbers that satisfy $ \rho(L_1) &gt; n/2 $ and $ \rho(L_2) &lt; (n+1)/2 $, and allow $ f(t, z) $ to grow superlinearly on the right half-plane. In order to estimate the rotation angle difference of solutions, we develop a system methodology of "tracking" the angle difference of solutions of the system $ z' = f(t, z) $ on each small interval on the given side under sign-varying conditions.</p></abstract>